New Constructions of Bent Functions Via 2-to-1 Mappings and Oval Polynomials

Bent functions are maximally nonlinear Boolean functions with critical applications in cryptography, combinatorics, coding theory, and sequence theory. Despite decades of research, their structural characterization remains incomplete, and existing constructions cover only a small fraction of all bent functions. This paper proposes a general framework for constructing bent functions via 2-to-1 mappings. Specifically, we achieve this by modifying the image sets of 2-to-1 mappings and investigating the Boolean functions supported on such sets. This framework enables the reconstruction of classical primary classes (e.g., the Maiorana-McFarland classM, Dillon’s class PSap, and Carlet-Mesnager’s class H) via 2-to-1 mappings with a specific structure. Capitalizing on this unified structure, we utilize oval polynomials to design 2-to-1 mappings and derive a large number of bent functions distinct from the main primary classes.